7.
Product and Process Comparisons
7.1.
Introduction
7.1.5.
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What is the relationship between a test and a confidence interval?
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| There is
correspondence between hypothesis testing and confidence intervals |
In general, for every test
of hypothesis there is an equivalent statement about whether the hypothesized
parameter value is included in a confidence interval. For example, consider
the previous example of linewidths where photomasks are tested to ensure that their linewidths have a mean of 500 micrometers. The null and alternative hypotheses are:
H0: mean linewidth = 500 micrometers
Ha: mean linewidth  500 micrometers
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| Hypothesis test for the mean |
For the test, the sample mean,  , is calculated from N linewidths chosen at random positions on each photomask. For the purpose of the test, it is assumed that the standard deviation,  , is known from a long history of this process. A test statistic is calculated from these sample statistics, and the null hypothesis is rejected if:
where  is a tabled value from the normal distribution. |
| Equivalent confidence interval |
With some algebra, it can be seen that the null hypothesis is rejected
if and only if the value 500 micrometers is not in the confidence interval
 .
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| Equivalent confidence interval |
In fact, all values bracketed by this interval would be accepted as null values for a given set of test data.
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